A086662 Stirling transform of Catalan numbers: Sum_{k=0..n} |Stirling1(n,k)|*C(2*k,k)/(k+1).

1, 1, 3, 13, 72, 481, 3745, 33209, 329868, 3624270, 43608474, 570008803, 8039735704, 121673027607, 1966231022067, 33786076421499, 615043147866660, 11822938288619344, 239298079351004608, 5086498410027323134, 113278368771499790136, 2637549737582063583274, 64082443707327038140602, 1621782672366231029685407

Offset: 0

Author

Vladeta Jovovic, Sep 12 2003

Keywords

• easy
• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000108, A008275, A064856.

Programs

Mathematica

CoefficientList[Series[(BesselI[0, 2*Log[1-x]] + BesselI[1, 2*Log[1-x]]) / (1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Mar 02 2014 *)
Table[Sum[Abs[StirlingS1[n,k]]*Binomial[2*k,k]/(k+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Mar 02 2014 *)

PARI

a(n)=sum(k=0,n, abs(stirling(n,k,1)) * binomial(2*k,k)/(k+1) ); \\ Joerg Arndt, Mar 02 2014

Formula

E.g.f.: hypergeom([1/2], [2], -4*log(1-x)) = 1/(1-x)^2*(BesselI(0, 2*log(1-x))+BesselI(1, 2*log(1-x))).

a(n)=(1/(2*pi))*int(product(x+k,k,0,n-1)*sqrt((4-x)/x),x,0,4) (moment representation). [Paul Barry, Jul 26 2010]